Situation

My PhD took place at Université de Sherbrooke, in the BISOUS team (in French), and at Inria Paris, in the SERENA team. It was supervised by Jean-Pierre Dussault (Sherbrooke) and Jean Charles Gilbert (Paris) since April 2021.

This PhD (since 2022) and the internship before it (2021) first focused on the globalization of the Levenberg-Marquardt algorithm applied to complementarity problems. When using a C-function, in particular the minimum, they can be expressed as a nonsmooth system, for which globalization of local methods such as the Newton-min algorithm is important.

The Bouligand differential of the minimum, which may intervene in such approaches is actually related to a problem from combinatorial geometry called hyperplane arrangements. We improved one of the main algorithms to obtain the chambers of an arrangement and showed some theoretical properties of arrangements through the prism of convex analysis and duality.

Via a relation between (centered) arrangements and zonotopes, particular polytopes, we were able to obtain theoretical properties describing strong stationarity (local optimality at first order) and that can reach a weakly stationary point (a differential contains zero).